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Proceedings of the 8th International Symposium on Mathematical Morphology,
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Locally nite spaces and the join operator
Erik Melin
Department of Mathematics, Uppsala University, Sweden
[email protected]
Abstract
Keywords:
The importance of digital geometry in image processing is well
documented. To understand global properties of digital spaces
and manifolds we need a solid understanding of local properties.
We shall study the join operator, which combines two topological
spaces into a new space. Under the natural assumption of local
niteness, we show that spaces can be uniquely decomposed as a
join of indecomposable spaces.
join operator, Alexandrov space, smallest-neighborhood space, locally nite space.
1. Introduction
Topological properties of digital images play an important role in image
processing. Much of the theoretical development has been motivated by the
needs in applications, for example digital Jordan curve theorems and the
theory of digitization. A classical survey is [11] by Kong and Rosenfeld.
Inspired by the new mathematical objects that have emerged from this
process, mathematicians have started to study digital geometry from a
more theoretical perspective, developing the theories in dierent directions.
Evako et al. [5, 6] considered, for example,
n-dimensional
digital surfaces
satisfying certain axioms. These surfaces were later considered by Daragon
et al. [4]. Khalimsky spaces as surfaces, embedded in spaces of higher dimension have been studied the author in [15].
We shall study, not digital spaces, but a tool that can be used in such
a study: the join operator.
Evako used an operation on directed graphs
called the join to aid the analysis. We will generalize this construction and
study its properties. In particular we show how a space can be decomposed
into indecomposable pieces put together by the join operator. We will give
conditions for uniqueness of this decomposition.
2. Digital spaces and the Khalimsky topology
We present here a mathematical background. The purpose is primarily to
introduce notation and formulate some results that we will need. A reader
not familiar with these concepts is recommended to take a look at, for
example, Kiselman's [9] lecture notes.
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Proceedings of the 8th International Symposium on Mathematical Morphology,
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2.1 Topology and smallest-neighborhood spaces
Not all topological spaces are reasonable digital spaces, but the class of
nite topological spaces is too small. It does not include
In every topological space, a
whereas an
arbitrary
nite
Zn .
intersection of open sets is open,
intersection of open sets need not be open.
If the
space is nite, however, there are only nitely many open sets, so nite
spaces fulll a stronger requirement: arbitrary intersections of open sets are
open.
Alexandrov [1] considers topological spaces, nite or not, that fulll
smallest-neighborhood
Alexandrov spaces, but this
this stronger requirement. We shall call such spaces
spaces.
Another name that is often used is
name has one disadvantage: it has already been used for spaces appearing
in dierential geometry.
Let
B
be a subset of a topological space
intersection of all closed sets containing
by
B.
We shall instead write
CX (B)
B.
X.
The
closure
of
B
is the
The closure is usually denoted
for the closure of
B
in
X.
This
notation allows us to specify in what space we consider the closure and is
NX dened below.
X as above, we dene NX (B) to be the intersection of all open sets containing B . In general NX (B) is not an open set, but
in a smallest-neighborhood space it is; NX (B) is the smallest neighborhood
containing B . If there is no danger of ambiguity, we will just write N (B)
and C (B) instead of NX (B) and CX (B). If x is a point in X , we dene
N (x) = N ({x}) and C (x) = C ({x}). Note that y ∈ N (x) if and only if
x ∈ C (y).
We have already seen that N (x) is the smallest neighborhood of x.
also a notation dual to
Using the same
B
and
Conversely, if every point of the space has a smallest neighborhood, then an
arbitrary intersection of open sets is open; hence this existence could have
been used as an alternative denition of a smallest-neighborhood space.
A point
x
point is called
pure,
open if N (x) = {x}, that is, if {x} is open. The
C (x) = {x}. If x is either open or closed it is called
called mixed.
is called
closed
otherwise it is
if
Adjacency and connectedness
A topological space
X
is called
connected
closed and open, are the empty set and
nent
if the only sets, which are both
X
itself.
A
connectivity compo-
(sometimes called a connected component) of a topological space is
a connected subspace which is maximal with respect to inclusion.
Two distinct points x and y in X are called adjacent if the subspace
{x, y} is connected. It is easy to check that x and y are adjacent if and only
y ∈ N (x) or x ∈ N (y). Another equivalent condition is y ∈ N (x) ∪ C (x).
The adjacency neighborhood of a point x in X is denoted ANX (x) and is
the set NX (x) ∪ CX (x). It is practical also to have a notation for the set of
points adjacent to a point, but not including it. Therefore the adjacency set
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x, denoted AX (x), is dened to be AX (x) = ANX (x) r {x}.
A (x) and AN(x).
A point adjacent to x is sometimes called a neighbor of x. This terminology, however, is somewhat dangerous since a neighbor of x need not be
in the smallest neighborhood of x.
in
X
of a point
Often, we just write
Separation axioms
Kolmogorov's separation axiom, also called the
two distinct points
x
and
y,
T0
axiom, states that given
there is an open set containing one of them
but not the other. An equivalent formulation is that
x=y
for every
Clearly any
T1/2
x
and
y.
The
space is also
T1/2
N (x) = N (y) implies
axiom states that all points are pure.
T0 .
The next separation axiom is the
T1
axiom.
It states that points are
closed. In a smallest-neighborhood space this implies that every set is closed
and hence that every set is open. Therefore, a smallest-neighborhood space
satisfying the
T1
axiom must have the discrete topology, and thus, is not
very interesting.
Duality
Since the open and closed sets in a smallest neighborhood space
X
satisfy
exactly the same axioms, there is a complete symmetry. Instead of calling
the open sets open, we may call them closed, and call the closed sets open.
Then we get a new smallest-neighborhood space, called the
we will denote by
X 0.
dual
of
X , which
The AlexandrovBirkho preorder
There is a correspondence between smallest-neighborhood spaces and par-
X be a smallest-neighborhood space and dene
x 4 y to hold if y ∈ N (x). We shall call this relation the Alexandrov
Birkho preorder. It was studied independently by Alexandrov [1] and by
tially preordered sets. Let
Birkho [3].
x is x 4 x)
x, y, z ∈ X , x 4 y and y 4 z imply x 4 z ). A relation
satisfying these conditions is called a preorder (or quasiorder ).
A preorder is an order if it in addition is anti-symmetric (for all x, y ∈ X ,
x 4 y and y 4 x imply x = y ). The AlexandrovBirkho preorder is an
order if and only the space is T0 . In conclusion, it would therefore be possible
The AlexandrovBirkho preorder is always reexive (for all
and transitive (for all
to formulate the results of this paper in the language of orders instead of
the language of topology.
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2.2 The Khalimsky topology
We shall construct a connected topology on
Z,
which was introduced by
Em Khalimsky (see Khalimsky et al. [8] and references there).
Pm = ]m − 1, m + 1[, and if m is an even
Pm = {m}. The family {Pm }m∈Z forms a partition of the
Euclidean line R and thus we may consider the quotient space. If we identify
each Pm with the integer it contains, we get the Khalimsky topology on Z.
We call this space the Khalimsky line. Since R is connected, the Khalimsky
If
m
is an odd integer, let
integer, let
line is a connected space.
It follows readily that an even point is closed and that an odd point is
open. In terms of smallest neighborhoods, we have
odd and
N (n) = {n − 1, n, n + 1}
if
n
N (m) = {m}
if
m
is
is even.
Perhaps this topology should instead be called the AlexandrovHopf
Khalimsky topology, since it appeared in an exercise [2, I:Paragraph 1:Exercice 4]. However, it was Khalimsky who realized that this topology was
useful in connection with digital geometry and studied it systematically.
Since this topology is also called the Khalimsky topology in the literature,
we will keep this name.
A dierent approach to digital spaces, using cellular complexes, was introduced independently by Herman and Webster [7] and by Kovalevsky [13].
The results of this article apply to such spaces, since they are topologically
equivalent to spaces of the type introduced in this article. See, for example,
Klette [10].
Khalimsky intervals and arcs
Let
a and b, a 6 b, be integers.
A
Khalimsky interval
is an interval
[a, b]∩Z of
integers with the topology induced from the Khalimsky line. We will denote
[a, b]Z and call a and b its endpoints. A Khalimsky arc in
X is a subspace that is homeomorphic to a Khalimsky
If any two points in X are the endpoints of a Khalimsky arc, we
X is Khalimsky arc-connected.
such an interval by
a topological space
interval.
say that
Theorem 1. A T0 smallest-neighborhood space is connected if and only if
it is Khalimsky arc-connected.
A proof can be found in [14, Theorem 11]. Slightly weaker is [8, Theorem 3.2c]. The theorem also follows from Lemma 20(b) in [12].
length of a Khalimsky arc, A, to be the number of points
L(A) = card A − 1. If a smallest-neighborhood space X
Let us dene the
in
is
A
T0
minus one,
and connected, Theorem 1 guarantees that length of the shortest arc
connecting
x
y in X is a nite number. This observation
ρX on X , which we call the arc metric.
and
dene a metric
ρX (x, y) = min(L(A); A ⊂ X
is a Khalimsky arc containing
66
allows us to
x
and
y).
Proceedings of the 8th International Symposium on Mathematical Morphology,
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The arc metric is dened on
the topology of
X
X,
but it is important to bear in mind that
is not the metric topology dened by
ρX .
The metric
topology is of course the discrete topology.
Examples of smallest-neighborhood spaces
We conclude this section with a few examples to indicate that the class of
smallest-neighborhood spaces and spaces based on the Khalimsky topology
is suciently rich to be worth studying.
Example 1.
The
Khalimsky plane is the Cartesian product of two KhalimKhalimsky n-space is Zn with the product topology.
sky lines and in general,
Points with all coordinates even are closed and points with all coordinates
odd are open. Points with both even and odd coordinates are mixed. It is
easy to check that
Example 2.
integer
A (p) = {x ∈ Zn ; kp − xk∞ = 1}
if
p
is pure.
Zm = Z/mZ for some even
Khalimsky circle. If m > 4, Zm is
We may consider a quotient space
m > 2.
Such a space is called a
a compact space that is locally homeomorphic to the Khalimsky line. (If
m
were odd, we would identify open and closed points, resulting in a space
with the indiscrete or chaotic topology, i.e., where the only open sets are
the empty set and the space itself.)
3. Locally nite and locally countable spaces
A topological space actually stored in a computer is nite. Nevertheless, it
is of theoretical importance to be able to treat innite spaces, like
Z2 ,
since
the existence of a boundary often tends to complicate matters.
On the other hand, spaces where a points can have an innite number
of neighbors seems less likely to appear in computer applications. In this
situation, not even the local information can be stored.
Denition 1.
A smallest-neighborhood space is called
ery point in it has a nite adjacency neighborhood.
countable adjacency neighborhood, it is called
locally nite
if ev-
If every point has a
locally countable.
We need to assume the axiom of choice (in fact the countable axiom of
choice, which states that from a
countable
collection of nonempty sets we
can select one element from each set, is sucient) to prove the following.
Proposition 1. Let X be a locally nite space. If X is connected, then X
is countable.
Proof.
If
X
is not
T0 ,
we consider the quotient space
identical neighborhoods have been identied.
X̃
X̃
is
T0
X̃ ,
where points with
and
X
is countable if
is countable. It is therefore sucient to prove the result for
67
T0
spaces.
Proceedings of the 8th International Symposium on Mathematical Morphology,
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x be any point in X . By induction based on local niteness, the
Bn (x) = {y; ∈ X; ρX (x, y) 6 n}, where ρX is the arc-metric on X
(here we use that the space is T0 ), is nite for every n ∈ N. Since X =
S∞
n=0 Bn (x), the result is true, since the countable axiom of choice implies
Let
ball
that a countable union of nite sets is countable.
In a similar way, we can also characterize countable smallest-neighborhood spaces.
Proposition 2. A smallest-neighborhood space X is countable if and only
if it is locally countable and has countably many connectivity components.
Proof.
It is clear that a countable space is locally countable and has count-
ably many components. The countable axiom of choice implies that countable unions of countable sets are countable. So if
X
is connected and locally
countable, a slightly modied version of the proof of Proposition 1 shows
that
X
is countable.
If
X
has countably many connectivity components
and each component is countable, then
X
is countable.
The following proposition states that the set of open points is dense in
a locally nite space (and by duality a corresponding result holds for the
closed points). It is obvious that the open points form the smallest dense
set.
Proposition 3. Let X be a smallest-neighborhood space and let S ⊂ X be
the set of open points in X and T be the set of closed points. If X is T0 and
locally nite, then X = C (S) = N (T ).
Proof.
y0
X . Let Y0 = N (y0 ). If Y0 is a singleton set, then y0 ∈ S .
Otherwise, for k > 0, choose yk+1 ∈ Yk r {yk } and let Yk+1 = N (yk+1 ).
Clearly Yk+1 ⊂ Yk and since X is T0 , it follows that yk 6∈ Yk+1 . Repeat the
construction above recursively until Yk is a singleton set, at most card(Y0 )−1
steps are needed. Note that yk is an open point and that y0 ∈ C (yk ). Since
y0 was arbitrarily chosen, we are done.
We will prove the st equality, the other follows by duality. Let
be any point in
The relation in the proposition need not hold if the space is not locally
nite. Consider the space
(non-trivial) open sets are given by
intervals
Z where the
]−∞, m]Z , m ∈ Z. This space
contains no open point.
−∞ and declare the open sets to
m ∈ Z ∪ {−∞}, then the point −∞ is open
C (−∞). Thus, nite neighborhoods are not
On the other hand, if we add the point
be intervals
[−∞, m]Z ,
where
and the whole space equals
necessary for the conclusion of the proposition.
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4. The join operator
A well-known way to combine two topological spaces,
the coproduct (disjoint union),
X
`
Y.
The pieces,
X and Y is to take
X and Y , are com-
pletely independent in this construction. We shall introduce another way
of combining two spaces, namely the join operator.
Denition 2.
Y,
denoted
Let
X
X ∨Y,
and
Y
be two topological spaces. The
˙ is declared
Y , where a subset A ⊂ X ∪Y
(i) A ∩ X is open in X and A ∩ Y = ∅, or
(ii) A ∩ X = X and A ∩ Y is open in Y .
and
(ii)
of
X
and
X
to be open if either
B ⊂ X ∨ Y is closed if and only
B ∩ X is closed in X and B ∩ Y = Y , or
B ∩ X = ∅ and B ∩ Y is closed in Y .
Note that a set
(i)
join
is a topological space over the disjoint set union of
if
While this denition makes sense for any topological space, it is a strange
operation on large spaces.
R ∨ S 1 , is a compact
cannot be T1 unless X or Y
circle,
For example, the join of the real line and the
space, which is
T0
T1 .
but not
In fact,
X ∨Y
is empty.
From now on, we shall only consider the join of smallest-neighborhood
X and Y
X ∨ Y is given by
NX∨Y (y) = NY (y) ∪ X if y ∈ Y . Ap-
spaces. In this case, the denition boils down to the following. If
are smallest-neighborhood spaces, then the topology of
NX∨Y (x) = NX (x)
if
x∈X
and
parently, the join of two smallest-neighborhood spaces is a smallest-neighborhood space.
This denition of the join is compatible with the join of
directed graphs, see [6, p. 111]. In terms of the AlexandrovBirkho preorder, every element of
X
X
in
is placed on top of
Y
Formally, the order, i.e.,
here denote
Ord(X ∨ Y )
is declared to be larger than any element of Y ;
X ∨ Y . This motivates also our notation X ∨ Y .
pairs (x, y) satisfying x < y , on X ∨ Y , which we
is
Ord(X ∨ Y ) = Ord(X) ∪ Ord(Y ) ∪ X × Y.
The join of two connected locally nite spaces need not be locally nite.
In fact, the join of two spaces is locally nite if and only if the spaces are
nite and locally countable if and only if both are countable.
In view of
Proposition 2, the join of two connected and locally countable spaces is
locally countable. When
Here
{p}
p
is a point, we write
p∨X
instead of
{p} ∨ X .
is the topological space with one point.
4.1 Basic properties
The following three properties in the next proposition are straightforward
to prove.
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Proposition 4. The join operator has the following properties for all
smallest-neighborhood spaces X, Y and Z .
(i) X = ∅ ∨ X = X ∨ ∅ (has a unity).
(ii) (X ∨ Y ) ∨ Z = X ∨ (Y ∨ Z) (is associative).
0
0
0
(iii) (X ∨ Y ) = Y ∨ X .
The following proposition lists some topological properties.
Proposition 5. Let X and Y be smallest-neighborhood spaces.
(i)
(ii)
(iii)
X ∨ Y is T0 if and only if X and Y are T0 .
X ∨ Y is compact if and only if Y is compact.
If X 6= ∅ and Y 6= ∅ then X ∨ Y is connected.
Proof. To prove (i), note that X is open in X ∨ Y . If x ∈ X and y ∈ Y ,
X is an open set containing x but not y . It follows that X ∨ Y can fail
to be T0 only for a pair of points in X or a pair of points in Y . But in this
then
case the equivalence is obvious.
Next we prove (ii). Assume rst that
is an open cover of
i ∈ I.
Then
Y
{Bi }i∈I
Y
is not compact and that
without a nite subcover. Let
is an open cover of
For the other direction, assume that
Y
X ∨Y
B i = Ai ∪ X
{Ai }i∈I
for each
without a nite subcover.
is compact and take an open cover,
{Bi }i∈I ,
of X ∨ Y . By restriction, it induces an open cover of Y with
Bi ∩ Y . But this cover has a nite subcover, {Bi ∩ Y }ni=1 , since Y
n
is compact. It follows that {Bi }i=1 is nite subcover of X ∨ Y since any Bi
where Bi ∩ Y 6= ∅ covers X .
To prove (iii), assume that x ∈ X and y ∈ Y . Since x ∈ N (y), it is
clear that x and y are adjacent. If a, b ∈ X , then {a, y, b} a connected set
for the same reason. In the same way {c, x, d} is connected if c, d ∈ Y .
elements
In fact all properties of the two proceeding propositions, except (iii) of
Proposition 4, are true for general topological spaces, not only for smallestneighborhood spaces.
4.2 Decomposable spaces
If
Z
Z = X ∨Y implies X = ∅ or Y = ∅, then the smallest-neighborhood space
is called indecomposable, otherwise Z is called decomposable. Note that a
locally nite decomposable smallest-neighborhood space is in fact nite.
Proposition 6. If a smallest-neighborhood space Z is decomposable, then
for every x, y ∈ Z there is a point z ∈ Z such that x, y ∈ AN(z). (If Z is
T0 , an equivalent and more comprehensible condition is that ρZ (x, y) 6 2.)
Proof.
The result is given by the proof of Proposition 5, Part (iii).
The converse implication is not true, as the following example shows.
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Example 3.
X = {a1 , a2 , b, c1 , c2 } be a set with 5 points, and equip it
N (a1 ) = {a1 }, N (a2 ) = {a2 }, N (b) = {a1 , b},
N (c1 ) = {a1 , a2 , b, c1 }, and N (c2 ) = {a1 , a2 , c2 }. It is easy to see that
ρ(x, y) 6 2 for any x, y ∈ X .
Suppose that X were decomposable, X = A ∨ C . We would necessarily
have a1 , a2 ∈ A since these points are open, and c1 , c2 ∈ C since these points
are closed. But b cannot be in A since b 6∈ N (c2 ) and b cannot be in C
since a2 6∈ N (b). Therefore, X is indecomposable.
Let
with a topology as follows:
We have the following uniqueness result for the decomposition.
Theorem 2. Let X be a smallest-neighborhood space. If X = Y ∨ Z and
X = Ỹ ∨ Z̃ , where Y and Ỹ are indecomposable and non-empty, then Y = Ỹ
and Z = Z̃ .
Proof. It is sucient to prove that Y = Ỹ . If Y 6= Ỹ , we may suppose that
Y r Ỹ is not empty and let p ∈ Y r Ỹ . Then Ỹ ⊂ Y , for if there were a
point q in Ỹ r Y , then q ∈ Z so that CX (q) ⊂ Z , since X = Y ∨ Z . But
by assumption p 6∈ Ỹ , so therefore p ∈ Z̃ . As q ∈ Ỹ and X = Ỹ ∨ Z̃ , this
implies p ∈ CX (q) ⊂ Z , which is a contradiction since p ∈ Y .
Dene B = Y r Ỹ , which is non-empty by assumption. Take two arbitrary points a ∈ Ỹ and b ∈ B . Note that b ∈ Z̃ . Hence a ∈ NX (b)
and thus also a ∈ NY (b). It follows that NY (b) = NB (b) ∪ Ỹ . If Ỹ and
B are equipped with the relative topology, we have Y = Ỹ ∨ B , so Y is
decomposable contrary to the assumption.
If a smallest-neighborhood space
X
is locally nite, then repeated use
of Theorem 2 together with associativity gives the following.
Corollary 1. If X is a locally nite smallest-neighborhood space, then X
can be written in a unique way as X = Y1 ∨ · · · ∨ Yn where each Yi is
indecomposable and non-empty.
Note that if
X
is decomposable so that
n > 1,
then
X
is necessarily
nite. While Theorem 2 is quite general, there is no universal cancellation
law; if
A∨X = B ∨X
we cannot conclude that
(which we will denote by
Example 4.
A ' B ),
A and B
are homeomorphic
as the following example shows.
N denote the set of natural numbers equipped with the
N (n) = {i ∈ N; i > n}. For a positive integer m, let
Nm = N ∩ [0, m], with the induced topology. It follows that N = Nm ∨ N
for every m, but Nm is homeomorphic to Nk only if m = k .
Let
topology given by
On the other hand, if
X
is locally nite, the unique decomposition of
Corollary 1 implies that both a right and a left cancellation law hold. In
fact, we may weaken the hypothesis slightly.
Theorem 3. Let X be a smallest-neighborhood space. Suppose there are
nitely many locally nite spaces Y1 , . . . , Yn so that X = Y1 ∨ · · · ∨ Yn . Then
for all smallest-neighborhood spaces A and B we have
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(i)
(ii)
X ∨ A ' X ∨ B implies A ' B ,
A ∨ X ' B ∨ X implies A ' B .
Note that
X
n=1
is locally nite only if
or if every
Yi
X
(and hence
itself ) is nite.
Proof.
Let Y
opening chain in Y to be
a nite sequence of pairwise distinct points (y0 , . . . , yn ) in Y such that
yi+1 ∈ NY (yi ) for 0 6 i 6 n. The number n is the called the length of
the open chain. We say that the chain starts in y0 . Let hY : Y → N ∪ {∞}
We shall prove (i).
The second claim follows by duality.
be any smallest-neighborhood space. Dene an
be dened by
hY (y) = sup(n;
there is an opening chain in
X , it
x ∈ X.
Y
of length
n
starting in
From the construction of
is straightforward to check that
a nite number for any
Furthermore, for any
a∈A
y).
hA∨X (x)
is
we have
hX∨A (a) > 1 + sup hX∨A (x)
(1)
x∈X
since every
x∈X
belongs to
NX∨A (a).
Furthermore,
sup hX∨A (x) = sup hX∨B (x),
x∈X
(2)
x∈X
hX (x) = hX∨Y (x) for any space Y and any x ∈ X .
φ : X ∨ A → X ∨ B be a homeomorphism. A chain is mapped to
a chain by φ, so we have the identity hX∨A (x) = hX∨B (φ(x)) for every
x ∈ X ∨ A. In view of (1) and (2) this implies that
since
Let
hX∨B (φ(a)) > 1 + sup hX∨B (x),
x∈X
for every
a ∈ A.
Hence
φ(X) = X
and
φ(A) = B ,
which proves (i).
5. Applications
We shall demonstrate how the tools we have developed can be used to give
a simple proof of a known result in digital topology, namely the characterization of neighborhoods in Khalimsky spaces (Evako et al. [6]). We start
with a consequence of Proposition 6.
Corollary 2. Let p ∈ Zn be pure. Then A (p) is indecomposable. If p is
closed, AN(p) = A (p) ∨ p and if p is open, AN(p) = p ∨ A (p).
Proof.
q ∈ A (p) then r = 2p − q
Zn ⊂ Rn ) also belongs to A (p). It is readily checked that
ρA (p) (q, r) = 4 if n > 2 (if n = 1, AZ (p) is not connected and the result immediate). Proposition 6 shows that A (p) is indecomposable. The
decomposition of AN(p) is straightforward.
For the rst property, notice that if
(as vectors in
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Rio de Janeiro, Brazil, Oct. 10 –13, 2007, MCT/INPE, v. 1, p. 63–74.
http://urlib.net/dpi.inpe.br/ismm@80/2007/03.19.21.57
If
p
is pure, it is easy to explicitly describe
AN(p).
We have
AN(p) = {x ∈ Zn ; kx − pk∞ 6 1},
3n − 1 points.
More generally, let q ∈ Z be any point with j
so that
p
is adjacent to
even coordinates and
k odd
coordinates. To simplify our notation, we note that there is a homeomor-
Zn , build from a translation q 7→ q +v , where v ∈ 2Zn , and permun
tation of coordinates, which takes q to the point q̃ = (0, . . . , 0, 1, . . . 1) ∈ Z ,
where there are j zeros and k ones (and j + k = n).
phism of
It follows that
N (q̃) = [−1, 1]j × {1}k
and that
C (q̃) = {0}j × [0, 2]k .
AN(q̃) = N (q̃) ∪ C (q̃), we see that q is adjacent to 3j + 3k − 2 points.
j
k
Let 0j denote the point (0, . . . , 0) ∈ Z and let 1k = (1, . . . , 1) ∈ Z .
It is easy to see that N (q̃) ' NZj (0j ) and it follows that N (q̃) r {q̃} is
homeomorphic to AZj (0j ), which is indecomposable by Corollary 2. By
a similar argument, C (q) r {q̃} is homeomorphic to AZk (1k ). Note that
A (00 ) = A (10 ) = ∅.
Since
It is straightforward to check that in any smallest-neighborhood space
X
and for any
x∈X
we have
A (x) ' (N (x) r {x}) ∨ (C (x) r {x})
and we obtain the following.
Proposition 7. Let q ∈ Zn . Then
AZn (q) ' AZj (0j ) ∨ AZk (1k ),
where j is the number of even coordinates in q and k is the number of odd
coordinates.
As stated from the outset, this result is known, and serves only as an
illustration of the formalism introduced.
6. Conclusion
We have studied the join operator, which takes two topological spaces and
combines them into a new space.
This operation is interesting primarily
for small spaces, viz. adjacency neighborhoods.
We have seen that if we
assume local niteness of the spaces involved, we can show that a space is
decomposed in a unique way into indecomposable spaces, and we have given
a criterion to recognize indecomposable spaces.
The machinery can be used to systematically investigate local properties
of digital topological spaces. Hopefully, this will lead to new insights into
the nature of such spaces.
73
Proceedings of the 8th International Symposium on Mathematical Morphology,
Rio de Janeiro, Brazil, Oct. 10 –13, 2007, MCT/INPE, v. 1, p. 63–74.
http://urlib.net/dpi.inpe.br/ismm@80/2007/03.19.21.57
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